In the context of linear parabolic equations, the Sobolev space $W^{1,q}\bigl([0,T], L^p(U) \bigr)$ appears all the time. Here, $U$ is some bounded region of $\mathbb{R}^n$ and $1<p,q<\infty$.
For any $f \in W^{1,q}\bigl([0,T], L^p(U) \bigr)$, standard PDE textbooks define $\partial_t f$ in terms of the Bochner integral.
Also, it is known that $W^{1,q}\bigl([0,T], L^p(U) \bigr) \subset C\bigl([0,T], L^p(U) \bigr)$ with properties of absolute continuity expressed in terms of Bochner integrals and so on.
Now, let us regard $f$ as a mapping from $[0,T]$ into $L^p(U)$. Then, I wonder if the temporal derivative $\partial_t f$ coincides with the "Fréchet derivative". That is, for almost every $t_0 \in [0,T]$, do we have
\begin{equation}
\frac{\lVert f(t_0+h)-f(t_0) +h(\partial_t f)(t_0) \rVert_{L^p(U)}}{\lvert h \rvert} \to 0
\end{equation}
as $h \to 0$
and $\partial_t f \in L^{q}\bigl([0,T], L^p(U) \bigr)$?
I think this is plausible but cannot justify myself.
